Lecture Combinatorial , Primal - Dual Approach to SDP . June 16 , 2011

Unless specified otherwise, all vectors in this lecture live in Rn, and all matrices are symmetric and live in Rn×n. For two vectors v,w, let v · w = ∑ i viwi denote their inner product, and v 0 indicate that all vi ≥ 0. For two matrices A and B, denote by A •B their inner product thinking of them as vectors in Rn2 , i.e. A • B = ∑ ij AijBij = Tr(A >B). Here Tr(·) denotes the trace of a matrix. A matrix A is positive semidefinite, denoted by A 0, if all its eigenvalues are non-negative. Equivalently, there are n vectors v1,v2, . . . ,vn such that Aij = vi · vj . We denote by A B the fact that A−B is positive semidefinite.